Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in $J$. Is there any characterization on a finite set of generators (probably a reduced Gröbner base) $G$ of $J$?

To rephrase my question, Is there a way to know when an ideal in $K[\underline{x}]$ has no monomials by just looking at a set of finitely many generators?